How should a backpacker fill out home address field in immigration/custom cards

Unless you've grown up on the streets you did have an address at some point in your life. Write down your last known address and forget about it. No one is going to check if you actually live there and nobody really cares. The field is just a formality in case they need your postal address for some reason. Whether or not you can actually receive correspondence at that address is not anyone's concern.

In some countries (e.g. Czech Republic) there is also the option to register yourself as a person without a residency, in which case your address would be the address of the local municipality office. This is the address that many homeless people have, as well as those who try to hide their real location from creditors, ex-husbands, etc.

All you need is an address. It is best though if it matches your other documents. For example, nearly every one who travels has a credit card and I do not know of any banks who would issue one without an address. Driver's licenses also have addresses in many country.

Using an address that matches helps you use these other documents to confirm your identity and will appear natural when immigration does a cross-check.

Now if you truly have no address written on any ID or bill or other official document, then I would use an old one first or some of close family. Should neither apply then you will have to be creative, perhaps workplace or a shared address used for deliveries.

The Where do you live? question is often asked at immigration, so I would much prefer to supply an address than leave it blank and try to explain that. Remember that they usually want to know that you can support yourself, so not having an address may raise question about employment.

If your passport (or even your national ID card) has a residential address listed (like the French passport/ID) then write down that address. That's what I do in the many countries that I travel to that require an address. The fact that this address may or may not be valid is beside the point. It's on the passport, it's official... :-)

If not, write whatever address you last had. It's not like they are going to check, unless you are from a country that is on their shit list. But in that case, you probably don't get visa-free entry, and would have to apply for a visa, where you'd have to provide an address...

The probability that their administration will want to contact you is very small -- and generally that would be while you are in-country, which is why they ask where you'll stay. And they can reach you at the border if they really want to talk to you.

Likewise, unless you're facing a very suspicious immigration officer (and your attitude justifies their suspicion), you won't have to prove your address. Never once, in close to 30 years of international travel, have I been asked once to prove my address at a border. And I used to carry a backpack and had long hair too! Now, not so much of course...

Bottom line, write down your last address and forget about it.